mancancercom revolving, as CD was, by the intervention of a groove and guide, then might the instrument be applied to overcome any given resistance R opposed to the motion of this piece CD by the constant pressure of its pivot upon that piece. The screw is applied under these circumstances in the common screw press. The piece A, fixed to the solid frame of the machine, contains a female screw whose thread corresponds to that of the male screw; this screw, when made to turn by means of a handle fixed across it, presses by the intervention of a pivot B, at its extremity, upon the surface of a solid piece EF moveable vertically, but prevented from turning with the screw by grooves receiving two vertical pieces, which serve it as guides, and form parts of the frame of the machine. The formulæ determined in Art. 251. for the preceding case of the application of the screw, obtain also in this case, if we assume 6.50. The loss of power due to the friction of the piece EF upon its guides will, however, in this calculation, be neglected; that expenditure is in all cases exceedingly small, the pressure upon the guides, whence their friction results, being itself but the result of the friction of the pivot B upon its bearings; and the former friction being therefore, in all cases, a quantity of two dimensions in respect to the coefficient of friction. If, instead of the lamina A (p. 354.) being fixed upon the convex surface of a solid cylinder, and B upon the concave surface of a hollow cylinder, the order be reversed, A being fixed upon the hollow and B on the solid cylinder, it is evident that the conditions of the equilibrium will remain the same, the male instead of the female screw being in this case made to progress in the direction of its length. If, however, the longitudinal motion of the male screw B (p. 354.) be, under these circumstances, arrested, and that screw thus become fixed, whilst the obstacle opposed to the longitudinal motion of the female screw A is removed, and that screw thus becomes free to revolve upon the male screw, and also to traverse it longitudinally, except in as far as the latter motion is opposed by a certain resist ance R, which the screw is intended, under these circumstances, to overcome; then will the combination assume the well known form of the screw and nut. To adapt the formulæ of Art. 251. to this case, o, must be made =0, and instead of assuming the friction upon the extremity of the screw (equation 311) to be that of a solid pivot, we must consider it as that of a hollow pivot, applying to it (by exactly the same process as in Art. (251.), the formulæ of Art. (177) instead of Art. (176). The DIFFERENTIAL SCREW. 253. In the combination of three inclined planes discussed in Art. 245., let the plane B be conceived of much greater width than is given to it in the figure (p. 347.), and let it then be conceived to be wrapped upon a convex cylindrical surface. Its two edges ab and cd will thus become the helices of two screws, having their threads of different inclinations wound round different portions of the same cylinder, as HPPHITHER THE PATHWAB represented in the accompanying figure, where the thread of one screw is seen winding upon the surface of a solid cylider from A to C, and the thread of another, having a diferent incrination, from D to B. Let, moreover, the planes A and C (p. 347.) be imagined to be wrapped round two borios cylindrical surfaces, of equal diameters with the above-mentioned solid cylinder, and contained within the solid pieces E and F, through which bollow cylinders AB passes. Two female screws will thus be generated within the pieces E and F, the belix of the one adapting itself to that of the male screw extending from A to C, and the helix of the other to that upon the male screw extending from D to B. If, then, the piece E be conceived to be fixed, and the piece F moveable in the direction of the length of the screw, but prevented from turning with it by the intervention of a guide, and if a pressure P, be applied at A to turn the screw AB, the action of this combination will be precisely analogous to that of the system of inclined planes discussed in Art. 245., and the conditions of the equilibrium precisely the same; so that the relation between the pressure P, applied to turn the screw (when estimated at the circumference of the thread) and that Py, which it may be made to overcome, are determined by equation (301), and its modulus by equation (302). The invention of the differential screw has been claimed by M. Prony, and by Mr. White of Manchester. A com. paratively small pressure may be made by means of it to yield a pressure enormously greater in magnitude.* It admits of numerous applications, and, among the rest, of that suggested in the preceding engraving. * It will be seen by reference to equation (301), that the working pressure P, depends for its amount, not upon the actual inclinations hy n of the threads, but on the difference of their inclinations; so that its amount may be enormously increased by making the threads nearly of the same inclination. Thus, neglecting friction, we have, by equation (301), cos. 4, C08. tam Pz=lisin. (1 in) '; which expression becomes exceedingly great when A, nearly equals 1z. CHE HUNTER'S SCREW. 254. If we conceive the plane B (p. 347.) to be divided by a horizontal line, and the upper part to be wrapped upon the inner or concave surface of a hollow cylinder, whilst the lower part is wrapped upon the outer or convex circumference of the same cylinder, thus generating the thread of a female screw within the cylinder, and a male screw without it; and if the plane C be then wrapped upon the convex surface of a solid cylinder just fitting the inside or concave surface of the above-mentioned hollow cylinder, and the plane A upon a concave cylindrical surface just capable of receiving and adapting itself to the outside or convex surface of that cylinder, the male screw thus generated adapting itself to the thread of the screw within the hollow cylinder, and the female screw to the thread of that without it; if, moreover, the female screw last mentioned be fixed, and the solid male screw be free to traverse in the direction of its length, but be prevented turning upon its axis by the intervention of a guide; if, lastly, a moving pressure or power be applied to turn the hollow screw, and a resistance be opposed to the longitudinal motion of the solid screw which is received into it ; then the combination will be obtained, which is represented in the preceding engraving, and which is well known as Mr. Hunter's screw, having been first described by that gentleman in the seventeenth volume of the Philosophical Transactions. The theory of this screw is identical with that of the preceding, the relation of its driving and working pressures is determined by equation (301), and its modulus by equation (302). every square unit of it, is represented by RD. Let Ar represent the exceedingly small thickness of such a ring whose radius is r, and which may therefore be conceived to represent the termination of the exceedingly thin cylindrical surface passing through the point p; the area of this ring is then represented by 2nrAr, and therefore the pressure upon it by P2xrAr, or by PirAr. Now this is evidently the pressure sustained by that elementary portion of the thread which passes through p, whose thickness is Ar, and which may be conceived to be generated by the enwrapping of a thin plane, whose inclination is , upon a cylinder whose radius is r; whence it follows (by equation 311) that the elementary pressure AP, which must be applied to the arm of the screw to overcome this portion of the resistance Pn, thus applied parallel to the axis upon an element of the thread, is represented by (PyrAr) (TM) sin.(r+.)cos. 93 + 8. tan.cz }; APi= (2RD)(a{ cos.(++03) + 3.can. whence, passing to the limit and integrating, we have R+D P, sin.(1 + i) cos. P32 f192RDa ) | cos. (1 + +P3)" 7 3po van.Y2 J " R-D Now sin.(o+0.)cos. 03 – tan..+ tan. cos. (1+0, +83) = 1-tan.P. tan. 03 — tan. (tan.P. + tan.03) tan.. + tan. (1-tan.Q, tan. 93) {1-tan.. tan.(P1+03)} – 3){1-tan. tan. (oto)T=tan.8, + tan.. + tan. (P1+03) tan. 2.. Neglecting dimensions of tan. &, and tan. 9, above the first*, * The integration is readily effected without this omission ; and it might be desirable so to effect it where the theory of wooden screws is under discussion, the limiting angle of resistance being, in respect to such screws, considerable. The length and complication of the resulting expression has caused the omission of it in the text. |